where \(s=8\times 10^{-3}\)  is a small constant for numerical stability. We train our model for \(p_{\xi }(x_{t-1}|x_t)\)  with the simplified loss proposed by [1]}, using a neural network \(\mathrm {nn}_\xi (x_t, t)\)  that predicts the noise \(\epsilon \sim \mathcal {N}(0, I)\)  present at a given timestep (rather than the denoised mean values themselves). To handle the periodic nature of angular values, we introduce a function to “wrap” values within the range \([-\pi , \pi )\) : \(w(x) = ((x + \pi ) \bmod 2\pi ) - \pi \) . We use \(w\)  to wrap a smooth L1 loss [2]} \(L_{w}\) , which behaves like L1 loss when error is high, and like an L2 loss when error is low; we set the transition between these two regimes at \(\beta _L=0.1\pi \) . While this loss is not as well-motivated as torsional losses used by [3]}, we find that it achieves strong empirical results.
\(d_w &= w \left( \epsilon - \mathrm {nn}_\xi \left( w \left(\sqrt{\bar{\alpha }_t}x_0 + \sqrt{1 - \bar{\alpha }_t} \epsilon \right), t \right) \right) \\L_w &= {\left\lbrace \begin{array}{ll}0.5 \frac{d_w^2}{\beta _L} & \text{if $|d_w| < \beta _L$} \\|d_w| - 0.5 \beta _L & \text{otherwise}\end{array}\right.}\) 
